DISSIPATIVE SOLUTIONS AND THE INCOMPRESSIBLE INVISCID LIMITS OF THE COMPRESSIBLE MAGNETOHYDRODYNAMIC SYSTEM IN UNBOUNDED DOMAINS

. We consider the compressible Navier-Stokes system coupled with the Maxwell equations governing the time evolution of the magnetic ﬁeld. We introduce a relative entropy functional along with the related concept of dissipative solution. As an application of the theory, we show that for small values of the Mach number and large Reynolds number, the global in time weak (dis-sipative) solutions converge to the ideal MHD system describing the motion of an incompressible, inviscid, and electrically conducting ﬂuid. The proof is based on frequency localized Strichartz estimates for the Neumann Laplacean on unbounded domains.

1. Introduction. We consider the motion of a compressible, viscous, and conducting fluid confined to an unbounded domain Ω ⊂ R 3 . The commonly accepted system of MHD equations, describing the time evolution of the mass density = (t, x), the fluid velocity u = u(t, x), and the magnetic field B = B(t, x) reads: 3) where the viscous stress S is given by Newton's law and where the transport coefficients µ, ν, and λ are positive constants. The system is supplemented by the no-stick boundary conditions for the velocity field: u · n| ∂Ω , [Sn] ⊗ n| ∂Ω = 0, (1.5)

EDUARD FEIREISL, ANTONIN NOVOTNY AND YONGZHONG SUN
together with the assumption that ∂Ω is a perfect conductor, specifically, B · n| ∂Ω = 0, curl(B) × n| ∂Ω = 0, (1.6) where n denotes the outer normal vector to ∂Ω. Moreover, since our model domain is unbounded, we prescribe the far field behavior: → > 0, u, B → 0 as |x| → ∞. (1.7) We are interested in the inviscid, incompressible limit of the above system, meaning in the situation when

9) and
ε 0. Note that we have deliberately omitted the bulk viscosity component in (1.4), and, similarly, we fix γ = 5 3 in (1.8), where both assumptions reflect the characteristic properties of a monoatomic gas relevant in the context of plasma motion. The methods we use adapt easily to the more general case Our aim is to study the asymptotic behavior of solutions { ε , u ε , B ε } for ε → 0. The same problem in the whole space case Ω = R N and in the periodic case, N = 2, 3 was studied by Jiang, Ju, and Li [21], [22]. In particular, they identified the limit problem -the ideal MHD system -in the form: div x v = 0, (1.10)
(1.12) In the presence of a physical boundary, the system (1.10 -1.12) must be supplemented by the relevant boundary conditions v · n| ∂Ω = 0, H · n| ∂Ω = 0, (1.13) and, in accordance with (1.7), v, H → 0 as |x| → ∞. (1.14) Our main goal is to extend the result of Jiang et al. [22] to fluids confined to a physical domain with a boundary. Since our method is based on dispersive estimates eliminating the effect of acoustic waves, we need the domain Ω ⊂ R 3 to be unbounded. In contrast with the viscous incompressible limit, where the absence of proper eigenvalues of the acoustic generator is sufficient (and necessary) to provide the desired result (see [12]), the inviscid incompressible limit requires certain uniformity of the decay of acoustic waves, typically provided by the Strichartz estimates if Ω = R 3 . The validity of Strichartz estimates imposes severe restrictions on the geometric properties of the domain. The crucial observation exploited in the present paper is that it is enough to show frequency localized Strichartz estimates (see Section 5) valid for a considerably larger class of domains. We believe that this observation may be of independent interest.
As a model example, we consider a perturbed half space in R 3 , specifically, The propagation of acoustic waves is described by the scaled wave equation (1.16) where Φ is called acoustic potential. Using the results of Edward and Pravica [9] on the spectral properties of the Neumann Laplacean ∆ N , we show that , which, scaled in time, yields the desired decay estimates for the acoustic potential Φ, see Section 5. Note that the Neumann Laplacean ∆ N in a pertubed half-space admits, in general, a sequence of resonances in the complex plain approaching asymptotically the essential spectrum (see Edward and Pravica [9, Corollary 1]); whence (1.17) is probably optimal.
Our approach is based on the concept of suitable weak or dissipative solutions for the system (1.1 -1.6) proposed, in the context of the Euler system by DiPerna and Lions, see [30]. The dissipative solutions, introduced for the barotropic Navier-Stokes system in [16] and for the full Navier-Stokes-Fourier system in [14], satisfy the so-called relative entropy inequality representing a suitable extension of the standard energy inequality to a class of suitable "test" functions. In particular, a dissipative solution coincides with a (hypothetical) strong solution as long as the latter exists, see [14]. The idea of exploiting the effect of mechanical energy dissipation imposed in continuum mechanics by the Second law of thermodynamics goes back to the seminal paper by Dafermos [4]. More recently, Berthelin and Vasseur [1], Desjardins [7], and Germain [19] adapted this approach to problems in fluid mechanics. Notably, Germain [19] proved the weak-strong uniqueness for the barotropic Navier-Stokes system in the class of "more regular" weak solutions.
The paper is organized as follows. In Section 2, we recall the standard definition of weak solutions to the compressible MHD system (1.1 -1.7) and introduce the relative entropy, together with the related concept of dissipative solutions. We also show that any finite-energy weak solution to (1.1 -1.7) is a dissipative solution, in particular, the property of weak-strong uniqueness holds for the compressible MHD system -a result that may be of independent interest, see Section 2, Theorem 2.1. With all the preliminary material at hand, we state our main result concerning the singular limit in Section 3. The rest of the paper is devoted to the proof of convergence of solutions of the primitive system (1.1 -1.7) to the target system (1.10 -1.14). Using the relative entropy inequality we establish the necessary uniform bounds on the family of solutions { ε , u ε , B ε } ε>0 in Section 4. Section 5 is devoted to acoustic waves, in particular, we show the key estimate (1.17). The proof of convergence is completed in Section 6 by means of another application of the relative entropy inequality, where the "ansatz" is inspired by the pioneering paper of Masmoudi [32]. Possible extensions as well as limitations of the method are briefly discussed in Section 7.
To conclude this introduction, we point out that our basic framework is the theory of weak solutions and its application to singular limits in the spirit of the work by Lions and Masmoudi [31], see also the survey by Masmoudi [33] and the references cited therein. We also mention the papers by Hu and Wang [20], Kukučka [28], Kwon and Trivisa [29], where the method is applied to the incompressible limits for the MHD system. There is an alternative and more classical approach based on strong solutions proposed in the seminal paper by Klainerman and Majda [26] and later adopted and developed by many authors, the complete list of which goes beyond the scope of the present contribution. The interested reader may consult the surveys by Danchin [5], Gallagher [18], or Schochet [34].
2. Weak and dissipative solutions. We start with the standard definition of weak solutions to the compressible MHD system (1.1 -1.7), supplemented with the initial conditions: cf. [8].
• The equation of continuity (1.1) is replaced by a family of integral identities for any τ ∈ [0, T ] and any test function ϕ ∈ C ∞ c ([0, T ] × Ω). • The weak formulation of the momentum equation (1.2) reads • The energy inequality For the weak solutions to exist globally in time, the initial data must satisfy certain compatibility conditions: and where the latter bound holds if, for instance, . Under the hypotheses (2.7), (2.8), the existence of global-in-time weak solutions to the compressible MHD system (1.1 -1.7), (2.1) can be shown by the methods developed in Lions [30] and [15], cf. also [8].
2.1. Relative entropies and dissipative solutions. Motivated by [16], we introduce the relative entropy in the form where U, b, and r are smooth satisfying (2.12) Our next goal is to derive a relation for where the first integral on the right-hand side may be controlled by the energy inequality (2.5). On the other hand, the remaining integrals can be computed explicitly by means of (2.2 -2.4), specifically, by taking |U| 2 and P (r) − P ( ), U, and b as test functions in (2.2), (2.3), and (2.4), respectively. After a bit tedious but straightforward manipulation, we obtain the relative entropy inequality: where the remainder R reads A trio , u, B is called a dissipative solution of the compressible MHD system (1.1 -1.7), (2.1) if the relative entropy inequality (2.13) holds for all test functions r, U, b satifying (2.10 -2.12).
It can be shown, by means of the methods developed in [16], that any weak solution of the compressible MHD system (1.1 -1.7) is a dissipative solution.

2.2.
Weak-strong uniqueness. Given the regularity and integrability properties of a weak solution , u, B, validity of the relative entropy inequality (2.13) can be extended to a larger class of test functions r, U, b. In particular, we can establish the weak-strong uniqueness property for the compressible MHD system. Following the arguments of [16], we suppose that , u, B is a weak and r, U, b a sufficiently smooth solution of the compressible MHD system, both emanating from the same initial data. Here "sufficiently smooth" means that the relative entropy inequality (2.13) holds for r, U, and b. In particular, the strong solutions considered in Theorem 2.1 below fall in this category. Similarly to [16], we want to use a Gronwall type argument to conclude that, in fact, ≡ r, u ≡ U, and B ≡ b as long as the smooth solution exists.
Let (r, U, b) be a smooth solution of the same system, belonging to the class for a certain m ≥ 3, 0 < r ≤ r ≤ r < ∞, and emanating from the same initial data.
Remark. Of course, the conclusion of Theorem 2.1 holds for a large class of domains, in particular, for bounded domains in R 3 with sufficiently smooth boundary. We refer to [16] for details.
In what follows, we use the following structural properties of the function E introduced in (2.6): For r ∈ [r, r] ⊂ (0, ∞), we have where c 1 , c 2 are positive constants depending only on r, r. In particular, relation (2.16) yields To prove Theorem 2.1, we use the equations satisfied by the smooth solution (r, U, b) to replace the time derivatives in the relative entropy inequality. In comparison with [16], we get the following extra terms in (2.14) resulting from the presence of the magnetic field: Obviously, the last two integrals on the right-hand side may be absorbed by the left-hand side of (2.13) by means of the Gronwall argument. As for the first integral, we have Following the line of arguments of Germain [19], we may control the first term by where we have used (2.15), (2.17). On the other hand, in accordance with (2.16), Consequently, by virtue of Hölder inequality, . Therefore we may conclude that Finally, by virtue of (2.17), we have in particular, we may use a generalized version of Korn's inequality to obtain Thus, combining the previous estimates with those obtained in [16, Section 4.1] we may infer that which, by means a straightforward application of Gronwall's lemma to (2.13), completes the proof of Theorem 2.1.
3. Singular limit, the main result. Before stating our main result, we briefly discuss the question of existence of solutions to the target problem (1.10 -1.14). Here, the standard framework for local existence of smooth solutions is the scale of Sobolev spaces W m,2 , specifically, where m > 5 2 is a positive integer. Local existence of solutions on bounded domains have been established by many authors, see Kozono [27], Secchi [35], among others. The local existence results are easy to extend on "large" domains, in particular the perturbed half-space, by the method of invading domains, specifically, by replacing Ω by Ω r , The problem then reduces to finding suitable a priori bounds yielding the existence time T max independent of the size of the invading domain, cf. also Kato and Lai [24] and the remarks in Kato's paper [23]. We recall that, unlike the Euler system, where solutions are regular in the 2D-physical space, the solutions of the ideal MHD system do not, or at least are not known to, enjoy the same property so the local solutions are relevant also in the 2D-setting.
Let H denote the standard Helmholtz projection onto the space of solenoidal functions in Ω, specifically, v = H[v] + ∇ x Ψ, where the gradient component is the unique solution of the Neumann problem We refer to the monograph by Galdi [17] concerning the construction and the basic properties of H.
We are ready to formulate the main result of the present paper: Let Ω ⊂ R 3 be a perturbed half-space specified in (1.15). Let the pressure p = p ε and the transport coefficients ν ε , λ ε satisfy (1.8), (1.9), with Suppose that { ε , u ε , B ε } ε>0 is a family of the global-in-time weak (dissipative) solutions to the compressible MHD system (1.1 -1.7), with the initial data for any 0 < T < T max .
The rest of the paper is devoted to the proof of Theorem 3.1.
for any ϕ ∈ C ∞ c (Ω) and a certain g ∈ L 2 (Ω) . Since we assume that ∂Ω is smooth, we have It can be shown, see Edward and Pravica [9], that −∆ N is a non-negative selfadjoint operator on the Hilbert space L 2 (Ω), with the essential spectrum [0, ∞). Moreover, by virtue of Rellich's lemma (cf. Eidus [10, Theorem 2.1]), the point spectrum of −∆ N is empty. Indeed, supposing that Thus a direct application of Rellich's lemma yieldsṽ(x) = 0 for |x| > R; whence, by means of the unique continuation principle, v ≡ 0 in Ω.

5.1.
Limiting absorption principle and local energy decay. Since the point spectrum of −∆ N is empty, the result of Dermenjian and Guillot [6] implies that −∆ N satisfies the limiting absorption principle (LAP), specifically, the cut-off resolvent operator can be extended as a bounded linear operator on L 2 (Ω) for δ → 0 and µ belonging to compact subintervals of (0, ∞). As observed by Edward and Pravica [9, Corollary 1], the norm of this extension need not be bounded uniformly for µ ∈ (0, ∞) for certain domains Ω.
As shown in [11, Section 5.5], the validity of LAP yields the local energy decay estimates in the form for any ϕ ∈ C ∞ c (Ω) and any G ∈ C ∞ c (0, ∞).

5.2.
Dispersive estimates for the free Laplacean. We recall the standard Strichartz estimates for the free Laplacean ∆ in R 3 , where H 1,2 denotes the homogeneous Sobolev space, see Keel and Tao [25], Strichartz [37]. In addition, the free Laplacean satisfies the local energy decay in the form

Frequency localized Strichartz estimates for the Neumann Laplacean.
Our goal is to show (1.17), specifically, To this end, we adapt the method developed by Burq [2], Smith and Sogge [36].
We decompose the function , 0 ≤ χ ≤ 1, χ even in x 3 , χ(x) = 1 for |x| ≤ R. Accordingly, w = w 1 + w 2 , where w 1 solves the homogeneous wave equation supplemented with the initial conditions Step 1. As a direct consequence of the standard Strichartz estimates (5.2), we get Step 2. Using Duhamel's formula, we obtain . At this stage, similarly to [2], we use the following result of of Christ and Kiselev [3]: Lemma 5.1. Let X and Y be Banach spaces and assume that K(t, s) is a continuous function taking its values in the space of bounded linear operators from X to Y . Set , where c 2 depends only on c 1 , p, and r.
We apply Lemma 5.1 to whence the desired conclusion (5.6) follows from the local energy decay estimates (5.3). As the norm of F is bounded in view of (5.1), we may infer that Step 3. Finally, since v = χU is compactly supported, we deduce from (5.1) combined with the standard elliptic regularity for −∆ N that while, by virtue of the standard energy estimates, where q < ∞ is the same as in (5.4). Interpolating (5.8), (5.9) and combining the result with (5.5), (5.7), we get (5.4). Finally, let us remark that (5.4) can be "strengthened" to 6. Convergence, part II. In this section, we complete the proof of Theorem 3.1 by showing the strong convergence of the velocity field claimed in (3.7), (3.8).
6.1. Oscillatory component of the velocity. We start by introducing the functions [s ε,δ , Φ ε,δ ] -the unique finite energy solution of the acoustic equation (1.16) emanating from the initial data where the [·] δ denotes a regularization defined as follows: The quantity ∇ x Φ ε,δ represents the (regularized) oscillatory component of the velocity field. Setting, for the sake of simplicity, a = 1 in (1.16), we recall the standard Duhamel's formula: In particular, we have the energy equality As a direct consequence of (6.5), regularity of the initial data, and the standard Sobolev embedding relation for any k = 0, 1, . . . Moreover, as shown in [13, Section 5.3], therefore the functions Φ ε,δ , s ε,δ decay fast for |x| → ∞ as soon as δ > 0 is fixed. Finally, it follows from the frequency localized Strichartz estimates (5.10) that where ω(ε, δ, k) → 0 as ε → 0 for any fixed δ > 0, k ≥ 0.

Estimates.
Similarly to the proof of weak-strong uniqueness, we conclude the proof of Theorem 3.1 by means of Gronwall type arguments. In what follows, we denote by g ε,δ = g ε,δ (t), h ε,δ = h ε,δ (t) generic functions such that We proceed in several steps: Step 1. We have Estimating the term in a similar manner we reduce the relative entropy inequality (6.9) to Step 2. Furthermore, where Ω ε ∇ x U ε,δ · (u ε − U ε,δ ) · (U ε,δ − u ε ) dx (6.14) ≤ Ω ∇ x U ε,δ (t, ·) L ∞ (Ω) E ε , u ε , B ε r ε,δ , U ε,δ , H dx can be "absorbed" by the left-hand side of (6.9) by means of a Gronwall type argument. More specifically, making use of (6.8), we have On the other hand, where, by virtue of the dispersive estimates (6.6), (6.8), the last three integrals vanish for ε → 0. As for the remaining two integrals, we have It follows from (4.11), (4.12), and the weak formulation of the equation of conti- weakly in L 1 (0, T ). Next, employing once more the dispersive estimates (6.6), (6.8), we obtain τ 0 Ω Finally, we get where, in accordance with (1.16), Similarly, Summarizing the previous discussion we may rewrite (6.13) in the form Step 3. We start with a simple observation where the former integral on the right-hand side tends to zero as ε → 0 as a consequence of the dispersive estimates (6.6), (6.8), while the latter reads where, by virtue of (6.6), (6.8), the last integral tends to zero for ε → 0.
Thus letting first ε → 0 and then δ → 0 completes the proof of Theorem 3.1.
7. Concluding remarks. The restrictions imposed on the "diffusion" coefficients ν ε ≈ ε a , λ ε ≈ ε b , a, b > 0, a ≤ 4 3 are probably not optimal. As a matter of fact, we do not need any restriction on λ ε but the restrictions imposed on ν ε are more restrictive than in [22].
In particular, this improves the result in [22]. Finally, we remark that the same result can be obtained on exterior domains as well as in other geometries that admit the relevant dispersive estimates for acoustic waves.